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Patterns in Cohomology of Moduli Spaces

This workshop will explore recent developments in the cohomology of many examples of moduli spaces, such as the classical moduli space of Riemann surfaces, moduli spaces of graphs or smooth manifolds, or related objects such as arithmetic groups. Talks by leading researchers will provide different perspectives on these spaces, and will explain new connections to homotopy theory and representation theory, new patterns in cohomology, and applications to arithmetic and geometry.

Speakers:  Alexander Berglund (Stockholm), Carel Faber (Utrecht) tbc, Benson Farb (Chicago), Ezra Getzler (Northwestern), Paul Gunnells (U Mass), Richard Hain (Duke), Alexander Kupers (Harvard), Eduard Looijenga (Utrecht and Tsinghua), Sam Payne (Austin), Dan Petersen (Stockholm), Andrew Putman (Notre Dame), Steven Sam (UCSD), Orsola Tommasi (Padova), Karen Vogtmann (Warwick), Craig Westerland (Minnesota)

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Venue: Mathematical Institute, University of Oxford

Water Waves and Related Fluid Models

The aim of this workshop is to discuss recent developments in the theory of water waves and related fluid models.  The main topics to be discussed are: local and global regularity, dynamical formation of singularities, and numerical methods.

In the past five years, new methods have emerged in the study of global solutions of quasilinear evolutions, inspired by the advances in semilinear theory.  The basic idea is to combine the classical energy and vector-fields methods with a new ingredient, namely refined analysis of the Duhamel formula using the Fourier transform method.  These new methods led to major progress in understanding the global dynamics of small-data solutions of many physically relevent Quasilinear models in 3 and 2 dimensions, including the construction of the first nontrivial global solutions in several classical water wave models.  At the same time, new methods have been developed to study the dynamics of large-data solutions, and the formation of certain types of interface singularities, such as the so-called splash singularity, wherein a locally smooth interface self-intersects in finite time.  The workshop will survey these recent developments and future research directions.

Speakers: Thomas Alazard (ENS), Peter Constantin (Princeton), Diego Cordoba (ICMAT), Daniel Coutand (Heriot Watt), Charles Fefferman (Princeton), Javier Gomez Serrano (Princeton), Alex Ionescu (Princeton), Alex Kiselev (Rice), David Lannes (Bordeaux), Victor Lie (Purdue), Nader Masmoudi (Courant), Benoit Pausader (Princeton), Fabio Pusateri (Princeton), Gregory Seregin (Oxford), Jalal Shatah (Courant), Steve Shkoller (UC Davis), Chongchun Zeng (Georgia Tech)

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Venue: Mathematical Institute, University of Oxford

New Frontiers in Probabilistic and Extremal Combinatorics

Speakers: Matija Bucic (Princeton), David Conlon (Caltech), Tim Gowers (Cambridge), Ben Green (Oxford), Annika Heckel (Uppsala), Oliver Janzer (Cambridge), Jeff Kahn (Rutgers), Peter Keevash (Oxford), Matt Kwan (IST), Imre Leader (Cambridge), Shoham Letzter (UCL), Dor Minzer (MIT), Richard Montgomery (Warwick), Will Perkins (GaTech), Alexey Pokrovskiy (UCL), Oliver Riordan (Oxford), Wojciech Samotij (Tel Aviv), Lisa Sauermann (Bonn), Mehtaab Sawhney (MIT), Asaf Shapira (Tel Aviv), Jacques Verstraete (UCSD)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organizer.

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Venue: Mathematical Institute, University of Oxford

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The P=W Conjecture in Non Abelian Hodge Theory

The P=W Conjecture in the Non Abelian Hodge Theory of compact Riemann surfaces and for the general complex linear group was formulated by M.A. de Cataldo, T. Hausel and L. Migliorini in 2008. It was proved in the special cases of rank 2 and subsequently of genus 2. It has been proved in all rank and genera in 2022 by two independent teams. The conjecture is still open for other complex reductive groups and for higher dimensional complex projective manifolds. Since its inception, the conjecture has inspired and motivated work in several related areas: Non Abelian Hodge Theory (also in positive and mixed characteristics) such as Hitchin systems and character varieties, mirror symmetry, hyperkahler geometry, geometric representation theory, Donaldson-Thomas invariants, cluster varieties.

This workshop aims at bringing together experts in these related areas with the purpose of bringing into focus the state of the art in the subject, sharpening the vision for the near future and laying the ground for new collaborations. 

Speakers: Ben Davison (Edinburgh), Camilla Felisetti (Modena and Reggio Emilia), Andres Fernandez Herrero (Penn), Andrew Harder (Lehigh), Victoria Hoskins (Radboud), Tasuki Kinjo (Kyoto), Emmanuel Letellier (Jussieu), Anton Mellit (Vienna), Tony Pantev (Penn), Fernando Rodriguez Villegas (ICTP), Olivier Schiffmann (Paris-Saclay), Junliang Shen (Yale), Olga Trapeznikova (IST), Qizheng Yin (Peking), Siqing Zhang (IAS), Zili Zhang (Tongji)

Registration: Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organiser. 

Details

Venue: Mathematical Institute, University of Oxford

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Schedule Abstracts

New Advances in the Langlands Program: Geometry and Arithmetic

The last decade has seen significant progress in our understanding of the arithmetic of the Langlands program over number fields. Some of the inter-related themes to be explored in this workshop include: the proof of new cases of Langlands reciprocity, linking Galois representations and automorphic forms; breakthroughs in understanding the geometry of Shimura varieties (particularly their analytic geometry over p-adic fields); the growing role of moduli spaces of Galois representations.

Speakers: George Boxer (Imperial), Matthew Emerton (Chicago), Tony Feng (Berkeley), Jessica Fintzen (Bonn), Toby Gee (Imperial), Ian Gleason (MPIM), Teruhisa Koshikawa (Kyoto), Si Ying Lee (Stanford), Jeffrey Manning (Imperial), Alice Pozzi (Bristol), Juan Esteban Rodriguez Camargo (Columbia), Peter Scholze (Bonn), Chris Skinner (Princeton), Naomi Sweeting (Harvard), Jack Thorne (Cambridge), Pol van Hoften (VU Amsterdam), Andrew Wiles (Oxford), Sarah Zerbes (ETH), Mingjia Zhang (Princeton)

This workshop is also supported by the ERC Starting Grant “P-adic arithmetic geometry, torsion classes, and modularity” (grant agreement No. 804176).

Registration: Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organizer.

Details

Venue: Mathematical Institute, University of Oxford

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Schedule Abstracts

Number Theory and Physics

This workshop will explore the interactions between number theory and physics, specifically focussing on four main points of contact:

  • L-function, Random Matrix Theory and Quantum Chaos 
  • Number Theory, Representation Theory and Donaldson-Thomas invariants
  • Feynman diagrams and multi-zeta values
  • The Geometric and Arithmetic Langlands Program

These topics, of current interest and activity, share a significant intersection with both number theory and physics and strongly influence and enrich one another. 

Each morning an expository talk will address one of these topics. The talks will be broad and expository, with an emphasis on the underlying theme of the workshop: the interplay between number theory and physics. Speakers are encouraged to include bold speculations and their vision for the future of the subject. The afternoon sessions will be devoted to more specialised talks.

The goal of the workshop is to bring together the four strands of the interaction between number theory and physics so that experts and scientists in general can glimpse the larger picture.

Speakers:  Manjul Bhargava (Princeton), Gaetan Borot (Geneve), Francis Brown (Jussieu), Philip Candelas (Oxford)), Sergei Gukov (Caltech), Andrew Hodges (Oxford), Jon Keating (Bristol), Maxim Kontsevich (IHES), Dirk Kreimer (Humboldt), Gérard Laumon (d’Orsay), Andrei Okounkov (Columbia), Marcus Reineke (Wuppertal), Zeev Rudnick (Tel Aviv), Edward Witten (IAS)

This is one of four concurrent workshops held in association with the 2013 Clay Research Conference.

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Venue: Mathematical Institute, University of Oxford

New Insights into Computational Intractability

Computational complexity theory deals with the central mystery of computation: What is feasible to compute, and what computational tasks will forever remain out of reach?  Although the main open questions (such as the Millennium Problem of P versus NP) continue to elude us, exciting progress has been reported on a number of fronts recently.  This workshop will bring together many leading researchers in the field of computational complexity theory, and provide a forum where they can share their perspectives on the new insights that have been obtained.

Speakers:  Scott Aaronson (MIT), Manindra Agrawal (IIT Kanpur), Eric Allender (Rutgers), Harry Buhrman (CWI Amsterdam), Peter Bürgisser (Paderborn), Stephen Cook (Toronto), Irit Dinur (Weizmann), Martin Dyer (Leeds), Lance Fortnow (Georgia Tech), Leslie Ann Goldberg (Liverpool), Paul Goldberg (Liverpool), Georg Gottlob (Oxford), Russel Impagliazzo (UCSD), Peter Jeavons (Oxford), Mark Jerrum (QMUL), Elias Koutsoupias (Oxford), Peter Bro Milterssen (Aarhus), Ketan Mulmuley (Chicago), Toniann Pitassi (Toronto), Rahul Santhanan (Edinburgh), Salil Vadhan (Harvard), Ryan Williams (Stanford)

This is one of four concurrent workshops held in association with the 2013 Clay  Research Conference.

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Venue: Mathematical Institute, University of Oxford

Quantum Mathematics and Computation

This inaugural research workshop of the Centre for Quantum Mathematics and Computation will seek to take stock of contemporary foundational ideas in mathematics, computer science and physics. Alongside invited talks by world experts, informal events will be held, such as problemsessions and research planning activities. Particular attention will be devoted to the organizing role played by logical structures, and how this illuminates our understanding of such diverse areas asquantum field theory, geometry, semantics of computation, and quantum information.

Speakers:  Samson Abramsky (Oxford), Steve Awodey (Carnegie Mellon), John Baez (UC Riverside), Alexander Beilinson (Chicago), Bob Coecke (Oxford),  Lucien Hardy (Perimeter), Martin Hyland (Cambridge),Yakov Kreminitzer (Oxford), Dana Scott (Carnegie Mellon), Steve Simon (Oxford), Vladimir Voevodsky (IAS), Anton Zeilinger (Vienna)

This is one of four concurrent workshops being held in association with the 2013 Clay Research Conference.

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Venue: Mathematical Institute, University of Oxford

The Navier-Stokes Equations and Related Topics

This three-day workshop will bring together world-leading experts in the field of mathematical hydrodynamics to discuss recent developments in and future directions of research in this very active area of applied analysis.

Topics will include the existence, uniqueness and regularity of the Navier-Stokes equations, Euler’s equations and other related hydrodynamics models, as well as numerical aspects. Recent developments in the mathematical study of incompressible fluid models will also be discussed, including, but not limited to:

  • progress on regularity of solutions to the Navier-Stokes equations and certain incompressible geophysical and polymeric fluid models
  • large-scale numerical simulations that shed some light on conjectured scenarios for the blow-up of solutions of the 3D Euler’s equations
  • progress on regularity of quasi-geostrophic equation with critical dissipation exponent
  • new approach to “rough solutions” of Euler via convex integration
  • the nearly complete solution of the Onsager conjecture about the energy preserving of the solutions to 3d Euler’s equations

The workshop will facilitate an exciting exchange of ideas, with new ideas emerging as a result of the interaction between the participants who have been chosen to represent the broad mathematical analysis community working on various aspects of hydrodynamics.

Speakers:  Diego Cordoba (ICMAT), Camillo De Lellis (Zürich), Yoshikazu Giga (Tokyo), Thomas Yuizhao Hou (Caltech), Alexander Kiselev (Wisconsin), Nader Masmoudi (NYU), Laure Saint-Raymond (ENS Paris), Vlad Vicol (Princeton), Sijue Wu (Michigan)

This is one of four concurrent workshops held in association with the 2013 Clay Research Conference.

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Venue: Mathematical Institute, University of Oxford

Analytic Number Theory

This workshop will cover all aspects of analytic number theory.  There have been a number of spectacular advances in recent years, on gaps between primes, mean-values of L-functions, Vinogradov’s mean-value theorem, rational points on algebraic varieties, and additive combinatorics, for example.

The aim of the workshop will be to bring together researchers in these various fields, to learn about the latest developments, facilitate interaction between different areas, promote collaborations, and to suggest future directions.

Invited speakers: Tim Browning (Bristol), Vorrapan Chandee (Burapha), Chantal David (Concordia), John Friedlander (Toronto), Leo Goldmakher (Williams), Andrew Granville (Montreal), Gergely Harcos (Renyi Institute), Adam Harper (Cambridge), Roger Heath-Brown (Oxford), Harald Helfgott (ENS), Emmanuel Kowalski (ETH Zurich), Dimitris Koukoulopoulos (Montreal), James Maynard (Montreal and Oxford), Lillian Pierce (Hausdorff Centre), Maksym Radziwill (IAS), Fernando Shao (Stanford), Kannan Soundararajan (Stanford), Trevor Wooley (Bristol)

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Venue: Mathematical Institute, University of Oxford

Symplectic Topology

The origins of symplectic topology lie in classical dynamics, and the search for periodic orbits of Hamiltonian systems.  It is now understood to arise naturally in algebraic geometry, in low-dimensional topology, in representation theory and in string theory.  Following seminal ideas of Gromov and Floer from the 1980s, several of the most powerful tools in symplectic topology revolve around invariants counting pseudoholomorphic curves.  An important theme in recent years has been that holomorphic curve invariants are not independent, but are bound together and governed by very rich algebraic structures, making connections to integrable systems, and to the theory of A-infinity and L-infinity algebras.  On the one hand, the algebraic structures make the invariants more powerful; on the other, these algebraic structures often underscore the connections to other fields, where similar structures arise with different origins.

The aim of this workshop is to bring together experts in various parts of holomorphic curve theory and to expose the latest developments in:

(i) Floer cohomology theory, the Fukaya category, and aspects of homological mirror symmetry
(ii) contact and Stein topology, and their connections to low-dimensional topology and to dynamics.

Recent areas of progress include flexibility results in Stein topology,  formulations and proofs of homological mirror symmetry, renewed ideas about flux and categorical dynamics, existence theorems for quasi-morphisms, sharp constraints on symplectic or Lagrangian embeddings, and relations to knot theory and finite type invariants.   Whereas much recent activity has been directed towards foundational issues, the workshop is intended to focus more on applications of the theory, particularly those where progress has been and is being made, and to highlight some of the new questions that progress raises.

Speakers: Mohammed Abouzaid (Columbia), Denis Auroux (Berkeley), Paul Biran (ETH Zurich), Vincent Colin (Nantes), Tobias Ekholm (Uppsala), Yasha Eliashberg (Stanford), Kenji Fukaya (Simons Center), Helmut Hofer (IAS), Michael Hutchings (Berkeley), Dusa McDuff (Columbia), Mark McLean (Stony Brook), Emmy Murphy (MIT), Tim Perutz (UT Austin), Leonid Polterovich (Tel Aviv), Paul Seidel (MIT), Nick Sheridan (Princeton)

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Venue: Mathematical Institute, University of Oxford

Functional Transcendence around Ax-Schanuel

Schanuel’s conjecture governs the transcendence properties of the exponential function.  In a differential field it is a theorem of Ax (1971).  Natural analogues for the uniformising maps of Shimura varieties, and related formulations, are of interest in Diophantine geometry, differential Galois theory, model theory, and complex geometry.  This workshop will gather participants from all these disciplines to share methods, results, and conjectures around this topic.

Speakers: Daniel Bertrand (Jussieu), Alexandru Buium (New Mexico), Philipp Habegger (Darmstadt), Jonathan Kirby (East Anglia), Bruno Klingler (Jussieu), Piotr Kowalski (Wroclaw), Angus Macintyre (Queen Mary London), David Masser (Basel), Ngaiming Mok (Hong Kong), Anand Pillay (Notre Dame), Michael Singer (North Carolina State), Emmanuel Ullmo (Paris Sud), Sai Kee Yeung (Purdue), Umberto Zannier (Pisa), Boris Zilber (Oxford)

Partially supported by EPSRC grant EP/J019232/1.

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Venue: Mathematical Institute, University of Oxford